Let $f\colon X \to \mathbb{A}^1_t$ be an affine flat morphism of finite type, and let $V = f^{-1}(0)$. Then, we obtain a morphism of log schemes $f\colon (X|V) \to (\mathbb{A}^1_t|0)$. In this article, we develop algorithmic tools to study the log-geometric properties of $f$ by means of a presentation \[Γ(X,\mathcal{O}_X) = \Bbbk[t,x_1,\ldots,x_n]/(f_1,\ldots,f_r).\] We obtain similar tools for projective flat morphisms when the homogeneous coordinate ring is given by generators and relations. We provide an implementation of our algorithms in Macaulay2. In a slightly different direction, we give some results on the sheaf $\mathcal{LS}_V$ of log smooth structures on a toroidal crossing scheme $(V,\mathcal{P},\barρ)$.
In this introduction, we motivate our results and give an overview. We discuss them in detail in Part I (which can and should be read as an extended introduction), and the proofs are provided in Part II.
We discuss the main result of this article.
Degenerating families of algebraic varieties. We fix an algebraically closed field k of characteristic 0. Let B/k be a smooth curve, and let f : X → B be a degenerating family of algebraic varieties. Let 0 ∈ B be a point, and suppose that V := f -1 (0) is the only singular fiber.
Example 1.1. We take B = Spec k[t] and X = Proj k[t, x, y, z, w]/(f 1 ) with deg(t) = 0 and deg(x) = deg(y) = deg(z) = deg(w) = 1, and with
This yields a degeneration f : X → B of a smooth quartic surface in P 3 to V = f -1 (0) = {xyzw = 0}. After restricting to a suitable open subset 0 ∈ B ⊆ B, we obtain a family f : X → B such that V = f -1 (0) is the only singular fiber. For an illustration of the central fiber, see [7,Fig. 1.2,Fig. 1.6]. ♢ Example 1.2. We take B = Spec k[t] and X = Proj k[t, x, y, z, u, v, w]/(f 1 , f 2 ) with deg(t) = 0 and the other variables of degree 1, and with
This yields a degeneration f : X → B of a smooth Fano threefold in P 5 . After restricting to a suitable open subset 0 ∈ B ⊆ B, we obtain a family f : X → B whose only singular fiber is V = f -1 (0). We will come back to these equations in Example 2. 21, where the reader can also find an illustration. ♢
We turn f into a morphism of logarithmic schemes f : (X|V ) → (B|0) by endowing source and target with the divisorial log structure, defined in the étale topology. For a description of this construction, see [23,§III.1.6]. In some cases, for example when f : X → B is semistable, this morphism is log smooth. Then, its formal properties are similar to those of smooth morphisms, and we can use log-geometric tools to analyze the degeneration. For example, when f : X → B is proper, the log Hodge-de Rham spectral sequence degenerates at E 1 , and the log Hodge sheaves are vector bundles [19]. We also have a logarithmic Gromov-Witten theory [15,1,2] that relates curve counts on V = f -1 (0) to curve counts on X t = f -1 (t) for t ̸ = 0.
In practice, f : (X|V ) → (B|0) is not log smooth for most degenerating families of algebraic varieties. For instance, neither the family in Example 1.1 nor the family in Example 1.2 is log smooth. However, similar to normal varieties, there is often a closed subset Z X ⊂ X of relative codimension ≥ 2 such that f : X \ Z X V \ Z X → (B|0) is log smooth. We say that f : (X|V ) → (B|0) is generically log smooth, and that Z X ⊂ X is the log singular locus. We recall details of this notion in Definition 2.2.
Our broader goal is to develop tools that do not only apply to log smooth families but to generically log smooth families. For example, in [9], we showed that the log Hodge-de Rham spectral sequence still degenerates at E 1 when the log singularities have a certain simple form. The log singularities in Example 1.1 have this form.
In contrast, the log singularities in Example 1.2 do not have this specific form. For more general log singularities like these, very little is known. For instance, we do not currently have a definition of the log de Rham complex for this family with the property that the log Hodge sheaves are vector bundles. The first step toward a broader understanding is to analyze concrete examples. In this article, we provide algorithmic tools to establish that a family f : (X|V ) → (B|0) given by generators and relations is indeed generically log smooth.
Computing the log singular locus. Suppose that we want to compute the log singular locus Z X ⊂ X from the equations. In Example 1.1, the central fiber V = {xyzw = 0} is a normal crossing scheme. By [7,Lemma 1.44], the morphism f : X → B is semistable in a point p ∈ V if and only if X is smooth in p. A direct computation yields Sing(X) ∩ V = Sing(V ) ∩ {x 4 + y 4 + z 4 + w 4 = 0}, a collection of 24 points, 4 on each of the six lines of D = Sing(V ). Then, f : X → B is semistable outside these 24 points, and hence generically log smooth with log singular locus Z X = Sing(X) ∩ V since f : X \ V → B \ {0} is smooth.
Let us try this approach in Example 1.2. The central fiber V has three irreducible components
Any two irreducible components intersect in a copy of P 2 . Concretely, let us write D(x) = V (x, w) ∩ V (x, y, z), D(y) = V (y, w) ∩ V (x, y, z), D(w) = V (x, w) ∩ V (y, w).
Then, D := D(x) ∪ D(y) ∪ D(w) is both the singular and the non-normal locus of V . Furthermore, all three irreducible components of V intersect in T ∼ = P1 . Now, V is not a normal crossing scheme along T . This is no problem as we only want to show log smoothness in codimension 1. Thus, we record T as part of the locus Z X ⊂ X where f : (X|V ) → (B|0) is not (necessarily) log smooth.
We consider D(x). Among points in D(x) \ T , the total space X is singular in Z(x) := D(x)∩{x 2 -uv = 0}. This is a curve, so we can add it to Z X .
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